02-16-2015, 02:09 PM

For a second-order ordinary differential equation,

y^('')+p(x)y^'+q(x)y=g(x).

(1)

Assume that linearly independent solutions y_1(x) and y_2(x) are known to the homogeneous equation

y^('')+p(x)y^'+q(x)y=0,

(2)

and seek v_1(x) and v_2(x) such that

y^* = v_1y_1+v_2y_2

(3)

y^('*) = (v_1^'y_1+v_2^'y_2)+(v_1y_1^'+v_2y_2^').

(4)

Now, impose the additional condition that

v_1^'y_1+v_2^'y_2=0

(5)

so that

y^('*)(x) = v_1y_1^'+v_2y_2^'

(6)

y^(''*)(x) = v_1^'y_1^'+v_2^'y_2^'+v_1y_1^('')+v_2y_2^('').

(7)

Plug y^*, y^*^', and y^*^('') back into the original equation to obtain

v_1(y_1^('')+py_1^'+qy_1)+v_2(y_2^('')+py_2^'+qy_2)+v_1^'y_1^'+v_2^'y_2^'=g(x),

(

which simplifies to

v_1^'y_1^'+v_2^'y_2^'=g(x).

(9)

Combing equations (◇) and (9) and simultaneously solving for v_1^' and v_2^' then gives

v_1^' = -(y_2g(x))/(W(x))

(10)

v_2^' = (y_1g(x))/(W(x)),

(11)

where

W(y_1,y_2)=W(x)=y_1y_2^'-y_2y_1^'

(12)

is the Wronskian, which is a function of x only, so these can be integrated directly to obtain

v_1 = -int(y_2g(x))/(W(x))dx

(13)

v_2 = int(y_1g(x))/(W(x))dx,

(14)

which can be plugged in to give the particular solution

y^*=v_1y_1+v_2y_2.

(15)

Generalizing to an nth degree ODE, let y_1, ..., y_n be the solutions to the homogeneous ODE and let v_1^'(x), ..., v_n^'(x) be chosen such that

{y_1v_1^'+y_2v_2^'+...+y_nv_n^'=0; y_1^'v_1^'+y_2^'v_2^'+...+y_n^'v_n^'=0; |; y_1^((n-1))v_1^'+y_2^((n-1))v_2^'+...+y_n^((n-1))v_n^'=g(x).

(16)

and the particular solution is then

y^*(x)=v_1(x)y_1(x)+...+v_n(x)y_n(x),